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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

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23 Aug 2015, 22:22

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Bunuel wrote:

For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

A. 30 B. 32 C. 46 D. 58 E. 64

Ans: D

Solution: 2^x + 2^y =2^30 there must be some value of as we know 2^30 only has 2 as its factor so if i take anything common from LHS of the equation the remaining part must be in form of 2 only. I can take common if and only if x=y; otherwise the remaining part inside the bracket will become odd for example lets say x= 14 and y = 16 then 2^14+2^16 = 2^14 (1+2^2)= 2^14 * 5 .. and this will happen for all the values of x and y where x is not equal to y so now=> the equation becomes 2^x + 2^x =2^30 because x=y 2^x (1+1)= 2^30 2^x * 2 = 2^30 x+1 = 30 x= 29 and y = 29 x+y = 58
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

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25 Aug 2015, 13:11

dkumar2012 wrote:

for example lets say x= 14 and y = 16 then 2^14+2^16 = 2^14 (1+2^2)= 2^14 * 5 .. and this will happen for all the values of x and y where x is not equal to y

This is a great solution. A little more detail for those wanting to see why all cases will not work (not just the one above)

2^x + 2^y = 2^x * (1+2^(y-x)) = 2^30

This means that 1+2^(y-x) has to be a multiple of 2 and this can only be achieved if 2^(y-x) = 1 = 2^0.

This problem is a good candidate for testing small numbers to find patterns. While your instincts may tell you to simply set x + y equal to 30, small numbers will show that you cannot simply set them equal. If you try \(2^x+2^y=2^6\), for example you can't find values for x + y = 6 that will set the sum equal to 64. The powers of 2 are:

2

4

8

16

32

64

The only pairing that will work is 32 + 32 = 64, meaning that you need \(2^5+2^5=2^6\), which should make sense: 2 times 2^5 will equal 2^6. So this example should teach you that you need to add two \(2^{29}\)s together to get to \(2^{30}\). So x and y are each 29, making the sum x + y equal to 58, answer choice D.
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

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08 Jan 2016, 09:08

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shasadou wrote:

For integers x and y, 2^x+2^y=2^30. What is the value of x+y?

A. 30 B. 32 C. 46 D. 58 E. 64

You are given 2^x+2^y=2^30 ---> in order to understand the question, try to see what values can y take if you start with fixed values for x ---> remember that x and y MUST be integers.

Thus, lets have x =1 ---> \(2^y=2^{30}-2\) ---> y can not be an integer. Try a bigger value for x.

x = 15 ---> \(2^y=2^{30}-2^{15}\) ---> \(2^y=2^{15}(2^{15}-1)\) , keep trying and you will see that there will always be a 1 inside the bracket except when you write the given expression as

\(2^x+2^y=2^{30}\) --->\(2^x+2^y=2*2^{29}\) ---> \(2^x+2^y=2^{29}+2^{29}\) ---> compare both sides of the equation to get x=y=29 and x+y = 58.

For integers x and y, 2^x+2^y=2^30. What is the value of x+y?

A. 30 B. 32 C. 46 D. 58 E. 64

Hi, here we are adding two different numbers which are power to base 2 resulting into another number which has a base of 2.. the logic is .. if we are adding 2^x and 2^y to get 2^z, only way is to get 2 out of the addition and that is possible only when x and y are same.. so 2^x+2^y=2^30..where x =y can be written as 2^x+2^x=2^30 2*2^x=2^30.. x+1=30 ..x=29.. so x+y=29+29=58
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

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16 Apr 2016, 18:28

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Here's a better way to solve the problem (suggested by Anthony Ritz - big thank you! ).

Step 1: factor out the given equation and see if there is a pattern \(2^x(1 + 2^{y-x}) = 2^{30}\) Hm, interesting. We know that \(2^x\) is always even, so \((1 + 2^{y-x})\) should also be even since \(2^{30}\) is even.

For \((1 + 2^{y-x})\) to be an even integer, \(2^{y-x}\) needs to be odd. What situation will \(2^{y-x}\) be an odd number? When \(2^{y-x} = 1\)! Hence,\(y-x = 0\) and consequently \(x=y\).

Step 2: plug in our findings to the equation \(2^x + 2^y = 2^x + 2^x = 2 * 2^x = 2^{1+x} = 2^{30}\) The last two parts of our equation tells us that \(1+x = 30\). In conclusion, \(x = y = 29\).

So 2^x +2^y = 2^n to get x+y you just do 2(n-1).
_________________

Took the Gmat and got a 520 after studying for 3 weeks with a fulltime job. Now taking it again, but with 6 weeks of prep time and a part time job. Studying every day is key, try to do at least 5 exercises a day.

Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

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05 Nov 2016, 22:34

Bunuel wrote:

For integers x and y, \(2^x + 2^y=2^{30}\). What is the value of \(x + y\)?

A. 30 B. 32 C. 46 D. 58 E. 64

Kudos for a correct solution.

Love this as a 700 level question. Easy if you keep your head on straight. Plug in 2^2 + 2^2= 8 = 2^3.... Plug in 2^3 +2^3= 16 = 2^4....As we can see the pattern shows that 2^29 + 2 ^29 will equal 2^30. Then we can add 29 + 29 and get 58.

Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

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